Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Dense ideals and cardinal arithmetic

From large cardinals we show the consistency of normal, fine, $\kappa$-complete $\lambda$-dense ideals on $\mathcal{P}_\kappa(\lambda)$ for successor $\kappa$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman

L-like Combinatorial Principles and Level by Level Equivalence

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like ” combinatorial principles. In particular, this model satisfies the following properties: 1. ♦δ holds for every successor and Mahlo cardinal δ. 2. There is a stationary subset S of the least supercompact cardinal κ0 such that for every δ ∈ S, ¤δ holds and δ carries a gap 1 morass. 3. A weak version of ¤δ holds for every infinite cardinal δ. 4. There is a locally defined well-ordering of the universe W, i.e., for all κ ≥ ℵ2 a regular cardinal, W H(κ+) is definable over the structure 〈H(κ+),∈ 〉 by a parameter free formula. ∗2000 Mathematics Subject Classifications: 03E35, 03E55. †Keywords: Supercompact cardinal, strongly compact cardinal, strong cardinal, level by level equivalence between strong compactness and supercompactness, diamond, square, morass, locally defined well-ordering. ‡The author’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. §The author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper. 1 The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Aspero ́ and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedman’s “outer model programme”.